Use of global interactions in efficient quantum circuit constructions Dmitri Maslov1, Yunseong Nam2;3 1 National Science Foundation, Arlington, VA 22230, USA 2 QuICS, University of Maryland, College Park, MD 20742, USA 3 IonQ Inc., College Park, MD 20740, USA [email protected], [email protected] January 1, 2018 Abstract In this paper we study the ways to use a global entangling operator to efficiently implement circuitry common to a selection of important quantum algorithms. In particular, we focus on the circuits composed with global Ising entangling gates and arbitrary addressable single-qubit gates. We show that under various circumstances the use of global operations can substantially improve the entangling gate count. Keywords: quantum circuits, quantum circuit optimization, global entangling operators, trapped ions, SIMD, quantum computing. 1 Introduction Trapped atomic ions [1] and superconducting circuits [2] are two examples of quantum information process- ing (QIP) approaches that have delivered small yet already universal and fully programmable machines. In superconducting circuits qubit interactions are enabled through custom designed electronic hardware involv- ing Josephson junctions and microwave resonators [2]. Different interactions can be controlled individually to invoke the two-qubit gates. A global coupling, however, would not necessarily be natural to such a system, due to the difficulty of placing and connecting O(n2) individual resonators in the same area as n qubits. This said, it is possible to couple Josephson junction qubits to a single resonator mode, thereby enabling global interactions [3]. In trapped ion QIP, on the other hand, global interactions are more naturally realized as an extension of common two-qubit gate interactions [4,5,6,7,8]. In fact, the ability to implement arbitrary selectable two-qubit interactions generally requires a higher level of control, with individually focused external fields addressing each qubit [1]. Given the ease of implementing a global interaction over these two leading QIP approaches, we consider the use of global entangling gates, particularly applied to the trapped ions technology. We note that the results are technology-independent and therefore apply to any QIP approach, so long as proper global entangling operations are constructible. One particular interaction available in the trapped ions approaches [1,7,8] to quantum computing is the so-called Molmer-Sorensen gate [9], also known as the XX coupling or Ising gate. To achieve computational arXiv:1707.06356v3 [quant-ph] 29 Dec 2017 universality, Molmer-Sorensen gate (either local addressable or global) is complemented by arbitrary single- qubit operations. These may come in different flavors, including the addressable R(θ; φ) rotations [1] of which at most two are needed to implement arbitrary single-qubit gate [10], or the addressable RZ rotation, which together with global RX and RY rotations also gives the single-qubit universality [7,8]. Depending on the specifics, the control apparatus may allow the application of an XX gate to a selectable pair of qubits [1], globally [4,5,6,7,8], or globally to a subset of qubits [11, 12]. We furthermore note that the existing control apparatus described in reference [1] allows the application of the global Molmer-Sorensen (GMS) gates [11], however, to date, this approach has not been studied in detail. In each case above, XX gate comes at a higher cost (expressed in terms of the duration and/or average fidelity) compared to the single-qubit gates. In this paper, we focus on minimizing the number of times an XX gate is called|be it addressable local or global, thereby targeting the most expensive resource in quantum computations using trapped ions QIP. 1 Specifically, we center our efforts on finding the instances of quantum computations that admit a more efficient implementation using global entangling gates compared to what may be accomplished using local entangling gates. Given that the control by global entangling operators applies a certain operation to multiple data, it can be thought of as a quantum analogue of classical SIMD (Single Instruction, Multiple Data) architecture. Our goal in this paper is thus to demonstrate practical advantages of quantum SIMD architecture beyond those examples already known. Previous work demonstrated how to implement the parity function (fan-in gate in our terminology) using a constant number of two global entangling pulses [13]. We revisit the implementation of fan-in in Subsection 3.1, since it is relevant to our more advanced constructions. Reference [14, Figure 5] shows a two-GMS gate construction of the number excitation operator used in quantum chemistry simulations [15, 16]. References [7, 17] study the ways to implement quantum algorithms efficiently on a trapped ion quantum computer with the two-qubit gates enabled by the global entangling operator, concentrating on the case featuring anywhere between two to four qubits. Reference [10] focuses on quantum circuit compiling in the scenario when local addressable two-qubit gates are available. Reference [18] revisits the two-GMS gate parity measurement implementation of [13] and reduces the number of global pulses needed to just one (this construction can be inferred from Fig.2), and shows how to measure the eigenvalue of a product of Pauli matrices using only a constant number of global entangling pulses. In contrast, here, we determine a set of important quantum circuits, focusing on the computations of arbitrary size, that can be accomplished using fewer entangling pulses in cases when global entangling control is available. The new circuits developed in our work include stabilizer circuits, Toffoli-4 gate, Toffoli-n gate, Quantum Fourier Transformation, and Quantum Fourier Adder circuits, thereby substantively extending the set of known efficient circuitry based on the global entangling pulse. The results are directly accessible for implementation over trapped ions approaches featuring global control, and make a case for mixed local/global entangling control. Computational universality of the control given by selectable two-qubit couplings and arbitrary single-qubit gates was the subject of an early foundational study establishing the upper bound of O(n34n) and the lower bound of Ω(4n) on the number of the CNOT gates required to implement an arbitrary unitary [19]. The upper bound was later improved to O(4n) in [20, 21], at which point it asymptotically met the lower bound, settling the question of asymptotically optimal control by the entangling CNOTs gates. For logical-level fault tolerant circuits one more step is needed|specifically, that of decomposing all gates into a discrete fault-tolerant library, such as the one given by the Clifford and T gates. With the CNOT being a Clifford gate, the remaining step on top of asymptotically optimal constructions of [20, 21] is to decompose arbitrary single-qubit unitaries into Clifford+T circuits. Euler angle decomposition may be used to express arbitrary single-qubit unitary as a circuit with no more than three axial rotations [22], and z-rotations can be synthesized optimally as single-qubit Clifford+T circuits [23, 24]. Given additional resources such as in-circuit measurement and classical feedback, even better solutions exist [25]. The upper bound of O(4n) on the number of CNOT gates [20, 21] gives rise to the upper bound of O(4n) on the number of GMS gates, since it can be easily established that a CNOT gate can be obtained using no more than constantly many GMS gates. Indeed, a 4-GMS implementation of the CNOT gate can be obtained by applying the two-GMS construction illustrated in Fig.1 to n qubits with χ = π=2, and then again to n−1 qubits with χ = −π=2, selecting one specific qubit-to-qubit interaction that remains active. With the use of the maximal size GMS gates, this may be a slightly larger construction, relying on Fig.8 to express smaller GMS gates in terms of the maximal size GMS gate, but one with constantly many GMS gates nonetheless. In most practical cases, one may desire to implement a specific well-structured computation, and those fre- quently come with known implementations relying on fewer than O(4n) entangling gates. Control by local addressable operations is clearly easier to work with as far as implementing quantum com- putations is concerned, since most quantum algorithms are expressed in terms of local operations. Secondly, (n−1)n the number of arbitrarily selectable two-qubit operations, 2 , for an n-qubit computation (recall that the XX coupling does not distinguish between gate's control and its target), is higher than 1, being the number of individual full-size global gates. Thirdly, an arbitrary circuit over two-qubit local control experiences only a constant factor blow up if needs be implemented as a circuit over global control (this is no more true if global control needs be expressed in terms of local control). These observations suggest that the local control is over- all more nimble when it comes to implementing arbitrary quantum algorithms. However, it is not always the case that the implementations using local addressable gates are more efficient compared to those over global 2 1 1 2 GMS3(−χ) 2 GMS4(χ) = XX(χ) 3 XX(χ) 3 XX(χ) 4 4 Figure 1: Example of the usefulness of global gates. GMS4 denotes a GMS gate defined according to (2), applied to all four qubits shown in the figure. GMS3 denotes a three-qubit GMS gate, applied to qubit numbers 1, 2, and 3. The common argument χ of the GMS gates specifies that all χij's are equal to χ. The XXij(χ) gate denotes a local XX gate, applied to qubits i and j with the angle χ, see (1). entangling operators. Indeed, it is known how to implement the 3-qubit Toffoli gate with only three size-3 GMS gates [7,8], whereas the best known implementation over two-qubit local addressable control requires five entangling gates [22].